Annular finite-time stability analysis and synthesis of stochastic linear time-varying systems

In this paper, we investigate some finite-time control problems for stochastic linear time-varying systems, described by an Itô type differential equation. In particular, the annular stochastic finite-time stability control problem is dealt with. In this context, it is required that the norm of the state variables remains bounded between an inner and an outer ellipsoid, during a finite-interval of time. The first contribution of the paper consists of two sufficient conditions for stability, which are derived by exploiting an approach based on time-varying quadratic Lyapunov functions. The first condition requires the solution of two generalised differential Lyapunov equations; the latter the existence of a feasible solution to a pair of differential linear matrix inequalities. As particular cases, we obtain sufficient conditions for stochastic finite-time stability (in this last case, the lower ellipsoid collapses to the origin). From the analysis conditions, a sufficient condition is derived for the annular stochastic finite-time stabilisation via both state and output feedback. Some numerical examples illustrate the application of the proposed methodology and show that our approach attains less conservative results than those obtainable with the existing literature. Moreover, an application of the annular stochastic finite-time stability approach to the stabilisation problem of a satellite with respect to the geomagnetic field is presented.